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受欢迎的三角函数 >

cos^2(x)+((1-sqrt(2))/2)cos(x)-(sqrt(2))/4 =0

解答

cos^{2} (x) + (\frac{1 - 2}{2}) cos (x) - \frac{2}{4} = 0

解答

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

+1

度数

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

求解步骤

cos^{2} (x) + (\frac{1 - 2}{2}) cos (x) - \frac{2}{4} = 0

用替代法求解

cos^{2} (x) + \frac{1 - 2}{2} cos (x) - \frac{2}{4} = 0

令

: cos (x) = u

u^{2} + \frac{1 - 2}{2} u - \frac{2}{4} = 0

u^{2} + \frac{1 - 2}{2} u - \frac{2}{4} = 0 : u = \frac{2}{2}, u = - \frac{1}{2}

u^{2} + \frac{1 - 2}{2} u - \frac{2}{4} = 0

找到

2, 4

的最小公倍数:

4

2, 4

最小公倍数 (LCM)

2

质因数分解:

2

2

2

是质数，因此无法因数分解

= 2

4

质因数分解:

2 \cdot 2

4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2

将每个因子乘以它在

2

或

4

中出现的最多次数

= 2 \cdot 2

数字相乘:

2 \cdot 2 = 4

= 4

乘以最小公倍数=

4

u^{2} \cdot 4 + \frac{1 - 2}{2} u \cdot 4 - \frac{2}{4} \cdot 4 = 0 \cdot 4

化简

4 u^{2} + 2 (1 - 2) u - 2 = 0

展开

4 u^{2} + 2 (1 - 2) u - 2 : 4 u^{2} + 2 u - 22 u - 2

4 u^{2} + 2 (1 - 2) u - 2

乘开

2 u (1 - 2) : 2 u - 22 u

2 u (1 - 2)

使用分配律:

a (b - c) = ab - a c

a = 2 u, b = 1, c = 2

= 2 u \cdot 1 - 2 u 2

= 2 \cdot 1 \cdot u - 22 u

数字相乘:

2 \cdot 1 = 2

= 2 u - 22 u

= 4 u^{2} + 2 u - 22 u - 2

4 u^{2} + 2 u - 22 u - 2 = 0

改写成标准形式

a x^{2} + b x + c = 0

4 u^{2} + (2 - 22) u - 2 = 0

使用求根公式求解

4 u^{2} + (2 - 22) u - 2 = 0

二次方程求根公式:

若

a = 4, b = 2 - 22, c = - 2

u_{1, 2} = \frac{- ( 2 - 2 2 ) \pm ( 2 - 2 2 ) ^{2} - 4 \cdot 4 ( - 2 )}{2 \cdot 4}

u_{1, 2} = \frac{- ( 2 - 2 2 ) \pm ( 2 - 2 2 ) ^{2} - 4 \cdot 4 ( - 2 )}{2 \cdot 4}

(2 - 22)^{2} - 4 \cdot 4 (- 2) = 2 + 22

(2 - 22)^{2} - 4 \cdot 4 (- 2)

使用法则

- (- a) = a

= (2 - 22)^{2} + 4 \cdot 42

数字相乘:

4 \cdot 4 = 16

= (2 - 22)^{2} + 162

乘开

(2 - 22)^{2} + 162 : 12 + 82

(2 - 22)^{2} + 162

(2 - 22)^{2} : 12 - 82

使用完全平方公式:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 2, b = 22

= 2^{2} - 2 \cdot 2 \cdot 22 + (22)^{2}

化简

2^{2} - 2 \cdot 2 \cdot 22 + (22)^{2} : 12 - 82

2^{2} - 2 \cdot 2 \cdot 22 + (22)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

2 \cdot 2 \cdot 22 = 82

2 \cdot 2 \cdot 22

数字相乘:

2 \cdot 2 \cdot 2 = 8

= 82

(22)^{2} = 8

(22)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (2)^{2}

(2)^{2} : 2

使用根式运算法则:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 2

= 2^{2} \cdot 2

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{2} \cdot 2 = 2^{2 + 1}

= 2^{2 + 1}

数字相加:

2 + 1 = 3

= 2^{3}

2^{3} = 8

= 8

= 4 - 82 + 8

数字相加:

4 + 8 = 12

= 12 - 82

= 12 - 82

= 12 - 82 + 162

同类项相加:

- 82 + 162 = 82

= 12 + 82

= 12 + 82

= 4 + 82 + 8

= 2 \cdot 2 + 82 + 8

= (2)^{2} (2)^{2} + 82 + (8)^{2}

8 = 22

8

8

质因数分解:

2^{3}

8

8

除以

28 = 4 \cdot 2

= 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

使用指数法则:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2

使用根式运算法则:

n ab = n a n b

= 2 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 22

= (2)^{2} (2)^{2} + 82 + (22)^{2}

222 \cdot 22 = 82

222 \cdot 22

数字相乘:

2 \cdot 2 = 4

= 4222

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

222 = 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{\frac{1}{2} + \frac{1}{2} + \frac{1}{2}}

= 4 \cdot 2^{\frac{1}{2} + \frac{1}{2} + \frac{1}{2}}

2^{\frac{1}{2} + \frac{1}{2} + \frac{1}{2}} = 22

2^{\frac{1}{2} + \frac{1}{2} + \frac{1}{2}}

合并分式

\frac{1}{2} + \frac{1}{2} + \frac{1}{2} : \frac{3}{2}

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + 1 + 1}{2}

数字相加:

1 + 1 + 1 = 3

= \frac{3}{2}

= 2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

使用指数法则:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

整理后得

= 22

= 4 \cdot 22

数字相乘:

4 \cdot 2 = 8

= 82

= (22)^{2} + 222 \cdot 22 + (22)^{2}

使用完全平方公式:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

(22)^{2} + 222 \cdot 22 + (22)^{2} = (22 + 22)^{2}

= (22 + 22)^{2}

使用根式运算法则:

n a^{n} = a

(22 + 22)^{2} = 22 + 22

= 22 + 22

使用根式运算法则:

a a = a

22 = 2

= 2 + 22

u_{1, 2} = \frac{- ( 2 - 2 2 ) \pm ( 2 + 2 2 )}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- ( 2 - 2 2 ) + 2 + 2 2}{2 \cdot 4}, u_{2} = \frac{- ( 2 - 2 2 ) - ( 2 + 2 2 )}{2 \cdot 4}

u = \frac{- ( 2 - 2 2 ) + 2 + 2 2}{2 \cdot 4} : \frac{2}{2}

\frac{- ( 2 - 2 2 ) + 2 + 2 2}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- ( 2 - 2 2 ) + 2 + 2 2}{8}

乘开

- (2 - 22) + 2 + 22 : 42

- (2 - 22) + 2 + 22

- (2 - 22) : - 2 + 22

- (2 - 22)

打开括号

= - (2) - (- 22)

使用加减运算法则

- (- a) = a, - (a) = - a

= - 2 + 22

= - 2 + 22 + 2 + 22

化简

- 2 + 22 + 2 + 22 : 42

- 2 + 22 + 2 + 22

同类项相加:

22 + 22 = 42

= - 2 + 42 + 2

- 2 + 2 = 0

= 42

= 42

= \frac{4 2}{8}

约分:

4

= \frac{2}{2}

u = \frac{- ( 2 - 2 2 ) - ( 2 + 2 2 )}{2 \cdot 4} : - \frac{1}{2}

\frac{- ( 2 - 2 2 ) - ( 2 + 2 2 )}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- ( 2 - 2 2 ) - ( 2 + 2 2 )}{8}

乘开

- (2 - 22) - (2 + 22) : - 4

- (2 - 22) - (2 + 22)

- (2 - 22) : - 2 + 22

- (2 - 22)

打开括号

= - (2) - (- 22)

使用加减运算法则

- (- a) = a, - (a) = - a

= - 2 + 22

= - 2 + 22 - (2 + 22)

- (2 + 22) : - 2 - 22

- (2 + 22)

打开括号

= - (2) - (22)

使用加减运算法则

+ (- a) = - a

= - 2 - 22

= - 2 + 22 - 2 - 22

化简

- 2 + 22 - 2 - 22 : - 4

- 2 + 22 - 2 - 22

同类项相加:

22 - 22 = 0

= - 2 - 2

数字相减:

- 2 - 2 = - 4

= - 4

= - 4

= \frac{- 4}{8}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4}{8}

约分:

4

= - \frac{1}{2}

二次方程组的解是:

u = \frac{2}{2}, u = - \frac{1}{2}

u = cos (x)

代回

cos (x) = \frac{2}{2}, cos (x) = - \frac{1}{2}

cos (x) = \frac{2}{2}, cos (x) = - \frac{1}{2}

cos (x) = \frac{2}{2} : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

cos (x) = \frac{2}{2}

cos (x) = \frac{2}{2}

的通解

cos (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (x) = - \frac{1}{2}

cos (x) = - \frac{1}{2}

的通解

cos (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

合并所有解

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

作图

流行的例子

tan^{2} (\frac{x}{2}) = - 2

sin^2(θ)-1=2sin(θ)

sin^{2} (θ) - 1 = 2 sin (θ)

- 3 sin^{2} (x) + 2 = 0

16sin(3x)cos(x)+8sqrt(3)sin(3x)-2cos(x)-sqrt(3)=0

16 sin (3 x) cos (x) + 83 sin (3 x) - 2 cos (x) - 3 = 0

tan (C) = 2.56